We consider a downlink wireless system consisting of one BS that serves certain eMBB and URLLC traffics simultaneously. The system bandwidth is partitioned into \(L\)  equally sized frequency resources, where each frequency resource is referred to as a resource element (RE). Each 12 REs constitute a resource block (RB) that is equivalent to 180 KHz. The time domain is divided into slots, also known as transmission time intervals (TTIs). The duration of each TTI is 1 ms. To support the low latency requirement of the URLLC traffic, each TTI is further divided into mini-slots, also known as small TTIs (sTTIs), where the duration of each sTTI is \(0.143\)  ms [1]}. The REs are assigned to the eMBB traffic at the beginning of each TTI, while the URLLC traffic arriving at each sTTI is directly transmitted in the next sTTI by puncturing the REs belonging to the eMBB load. Each URLLC packet is divided into blocks of \(\zeta \) -symbols, with \(\zeta \ge 1\) , and it is allocated within one sTTI. Each eMBB receiver is assumed to decode its received data without knowledge of the punctured resources, i.e, each eMBB receiver is assumed to be unaware of the punctured resources at the transmission. Accordingly, the puncturing overhead is reduced. Based on this assumption, each eMBB receiver decodes its received data according to its decoder. Let \(m\in \lbrace 2, 4, ...,{M}\rbrace \)  denote the order of the quadrature amplitude modulation (QAM) scheme adopted for the eMBB traffic with symbol error probability \(P_m(\gamma _{\rm e})\) , where \(\gamma _{\rm e}\)  denote the received eMBB signal-to-noise ratio (SNR). In addition, let \(n\in \lbrace 2, 4, ...,{N}\rbrace \)  denote the order of the QAM scheme adopted for the URLLC traffic and let \( \epsilon _{\rm u} \)  denote its target symbol error probability.By definition, both \(m\)  and \(n\)  are powers of 2. Practically, the URLLC modulation order \(n\)  is low due to the following reasons: 1) the lack of accurate channel estimation due to latency constraints; 2) the URLLC traffic is assumed to be small, so the achievable capacity follows the short-block regime; and 3) the high reliability constraint of the URLLC traffic, which means very low SER. We list in Table REF  most of the variables used in the analysis throughout the paper.
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